Viewing Eq.(7.2) in the frequency domain, the ideal differentiator transfer-function is , which can be viewed as the Laplace transform of the operator (left-hand side of Eq.(7.2)). Moving to the right-hand side, the z transform of the first-order difference operator is . Thus, in the frequency domain, the finite-difference approximation may be performed by making the substitution
in any continuous-time transfer function (Laplace transform of an integro-differential operator) to obtain a discrete-time transfer function (z transform of a finite-difference operator).
The inverse of substitution Eq.(7.3) is
Delay Operator Notation